Inequalities for exponential sums

We study the classes E-n := {f : f(t) = Sigma(n)(j=1) a(j)e(lambda jt), a(j), lambda(j) is an element of C}, E-n(+) := {f : f(t) = Sigma(n)(j=1) a(j)e(lambda jt), a(j), lambda(j) is an element of C, Re(lambda(j)) >= 0}, E-n(-) := {f : f(t) = Sigma(n)(j=1) a(j)e(lambda jt), a(j), lambda(j) is an element of C, Re(lambda(j)) <= 0}, and T-n := {f : f(t) = Sigma(n)(j=1) a(j)e(i lambda jt), a(j) is an element of C, lambda(1) < lambda(2) < . . . < lambda(n)}. A highlight of this paper is the asymptotically sharp inequality broken vertical bar f(0)broken vertical bar <= (1 + epsilon(n))3n broken vertical bar broken vertical bar f(t)e(-9nt/2) broken vertical bar broken vertical bar L-2[0,1], f is an element of T-n, where epsilon(n) converges to 0 rapidly as n tends to infinity. The inequality sup(0 not equivalent to f)is an element of Tn vertical bar f(0)vertical bar/vertical bar vertical bar f vertical bar vertical bar L-2[0,1] >= n is also established. Our results improve an old result due to Halasz and a recent result due to Kos. We prove several other related order-sharp results in this paper.